Download Allocation of risks and equilibrium in markets with finitely many traders

Allocation of risks and equilibrium in markets
with finitely many traders
Christian Burgert and Ludger Rüschendorf
University of Freiburg
Abstract
The optimal risk allocation problem, equivalently the optimal risk sharing problem,
in a market with n traders endowed with risk measure %1 , . . . , %n is a classical problem
in insurance and mathematical finance. This problem however makes only sense under
a condition motivated from game theory which is called Pareto equilibrium. There are
many situations of practical interest, where this condition does not hold. This is the
case if the risk measures are based on essential different views towards risk. In this
paper we introduce and analyze a meaningful extension of the optimal risk allocation
(risk sharing) problem without assuming the equilibrium condition. The main point of
this is to introduce a suitable and well motivated restriction on the class of admissible
allocations which prevents effects of artificial ‘risk arbitrage’. As a result we obtain
a new coherent risk measure which describes the inherent risk which remains after
using admissible risk exchange in an optimal way.
1
Introduction
In this paper we consider the problem of allocation of risk and characterizing equilibrium
in a market with n traders endowed with risk measures %1 , . . . , %n . A market is defined
to be in a Pareto equilibrium if in a balance of supply and demand it is not possible to
lower the risk of some traders without increasing the risk of some other traders. Based on
the duality theory of linear programming Heath and Ku (2004) characterized the Pareto
equilibrium condition in terms of the representing scenarios of the risk measures for the
case of a finite space Ω of possible outcomes.
In the first part of this paper we give a new derivation of the characterization of Pareto
equilibrum of Heath and Ku (2004) based on properties of related risk measures derived
from %1 , . . . , %n . In particular the infimal convolution is naturally associated to this problem.
We also give an extension to the case of incomplete markets, where trading and allocation
is possible only in linear subsets.
The optimal allocation of risk and the construction of reinsurance treaties is a classical
problem in insurance and is of considerable practical and theoretical interest. It has a long
history in the insurance literature going back to the construction of linear reinsurance
treaties based upon minimizing individual and aggregate variance of risk. (For references
see Seal (1969).)
In a series of important papers Borch (1960a, 1960b, 1962), Du Mouchel (1968), and
Gerber (1978) showed that based on utility functions Pareto optimal risk exchanges can be
characterized and in many cases lead to familiar linear quota-sharing of the total pooled
losses or to stop loss contracts and to mixtures of both. Solutions are however typically not
uniquely determined which may lead to the necessity to arrange substantial side payments
in order to make these solutions acceptable. In several papers authors have added game
theoretic considerations or additional concepts (like the concept of fairness) to arrive at
a specific element in the set of Pareto optimal rules (see Borch (1960b), Lemaire (1977),
Bühlmann and Jewell (1979)).
2
C. Burgert, L. Rüschendorf
Since risk pools redistribute only actual losses and possibly the associated premiums
but not the individual wealth of the company it is natural to include side constraints in
the exchange protocol of the form Yi ≥ Ai for the components Yi of the allocation and
some constant or random bounds Ai , to limit negative charges or payouts of company i.
Similarly also upper constraints of the form Yi ≤ Ai + Bi have been introduced to protect
the liquidity of the individual companies. The importance of side constraints has been
suggested by Borch (1968) and has formally been introduced and applied in Gerber (1978,
1979).
Several authors have extended the framework to include the presence of background
risk and have considered the allocation problem also in the context of financial risks (see
Leland (1980), Chavallier and Müller (1994), and Barrieu and El Karoui (2004, 2005),
Dana and Scarsini (2005), Chateauneuf, Dana, and Tallon (2000), Denault (2001) and
references therein). Also more general types of risk measures (distortion type, coherent,
convex, comonotone risk measures) have been considered for the allocation problem. For the
background literature on risk measures and their applications to finance and insurance we
refer to Deprez and Gerber (1985), Kaas, Goovaerts, Dhaene, and Denuit (2001), Delbaen
(2000), Delbaen (2002), and Föllmer and Schied (2004).
Our present paper is based on these developments. We consider the risk allocation
problem for a market where %i are coherent risk measures. This is the frame for which in
the paper of Heath and Ku (2004) the Pareto equilibrium was characterized (even if not
explicitely stated in that paper). In Remark 2.9 we comment on extensions of our results
to the more general case of convex risk measures. We show that the general formulation
of the optimal allocation problem in the sense of minimizing the sum of risks is only well
defined when the Pareto equilibrium condition holds. The main new part of this paper
is concerned with the allocation problem in the case that the equilibrium condition does
not hold. In this case the above formulation of the optimal allocation problem leads to
inconsistencies. We introduce a suitable class of restrictions on the set of allocations which
we call admissible allocations and consider the problem of optimal allocations with respect
to this restricted class. In comparison to the constraints as dealt with in Gerber (1978) we
postulate essentially constraints on the compensation structure of the form |Xi | ≤ |X| for
the allocation Xi motivated as above to limit negative charges or payouts and to protect the
liquidity. The bounds depend on the absolute size of the total risk X. From a mathematical
point of view our side constraints are connected with a similar idea in portfolio theory,
where one considers (lower bounded) admissible strategies in order to exclude strategies
which allow arbitrage. As consequence we obtain a new coherent risk measure – called the
coherent admissible
Pn infimal convolution risk measure – which describes the optimal total
admissible risk i=1 %i (Xi ) in the market. We could call this part of the risk the inherent
risk of the allocation problem, which remains even after optimally allocating the risk to the
n traders. The risk measure can be characterized as the largest coherent risk measure % such
that % ≤ min %i (Theorem 3.5). This result justifies our choice of restrictions as minimal
i
kind of restrictions leading to a senseful optimal allocation problem. Based on a general
version of the minimax theorem we are able to derive a simplified dual representation of
this risk measure (Theorem 3.1).
The risk sharing problem is a problem where the traders minimize the total risk by
some kind of exchange contracts. This can be considered as an ‘optimistic attitude’ towards
risk. It aims to construct an optimal admissible exchange which is typical for insurance
and reinsurance contracts. In the final part of our paper we consider the opposite view
from the perspective of a regulatory agent in a financial market who takes care that the
individual agents (traders)Phave enough capital reserves to cover their part of the risk
n
Xi in any allocation X = i=1 Xi to the n traders. The regulatory agent considers any
possible (admissible) allocation and determines the total risk in the worst case which is
the necessary total capital reserve. Therefore, we describe this situation as a situation as a
‘cautious risk attitude’. Again as a result we obtain a new coherent risk measure describing
the worst case total admissible risk.
Allocation of risks and equilibrium in markets with finitely many traders
3
Risk measures have a long tradition in the actuarial literature – denoted there as premium calculation principles. They also received considerable attention in the financial
mathematics literature more recently. Here a major need for risk measures is related to
pricing in incomplete models. Complete hedging of a claim is generally not possible but
even after hedging there remains a risky position. Thus the price of the claim depends on
the price of the hedging portfolio but also on the attitude of the agent (trader) towards
risk. There are essentially two ways to define risk measures. One way is to pose axioms on
a risk functional on the set of all risks (random variables). A second way is more economically motivated and based on the preference structure of the decision maker. These two
ways are essentially two equivalent ways of describing risk measures. For a presentation of
these views and a decription of their connections we refer to the recent informative survey
in Denuit et al. (2006).
In the final part of the introduction we give a short review of the basic notions on
coherent risk measures needed throughout the paper. A coherent risk measure % on a general
probability space (Ω, A, P ), i.e. % : L∞ (P ) → IR is a monotone, translation invariant,
subadditive homogeneous functional (see Delbaen (2002)). The associated acceptance set
A = A% is given by
A% = {X ∈ L∞ (P ); %(X) ≤ 0}.
(1.1)
% is a coherent risk measure if and only if A% is a monotone convex cone and inf{m ∈
IR; m ∈ A% } > −∞. Further, the following basic relation holds
%(X) = inf{m ∈ IR; X + m ∈ A% }
(1.2)
and (1.2) allows to define coherent risk measures from convex, monotone cones A. The
following representation result in terms of scenario sets P ⊂ ba(P ) – the finitely additive
measures absolutely continuous to P – is essential: % is a coherent risk measure if and only
if there exists a convex σ(ba(P ), L∞ (P ))-closed set P ⊂ ba(P ) such that
%(X) = sup EQ (−X),
∀X ∈ L∞ (P ).
(1.3)
Q∈P
Further, the representation scenario set P can be chosen in the class of probability measures
absolutely continuous to P if and only if any of the following equivalent conditions hold:
a) A = {X ∈ L∞ (P ); %(X) ≤ 0} is σ(L∞ (P ), L1 (P )) closed
(1.4)
b) % has the Fatou property, i.e. for any uniformly bounded sequence (Xn ) of random
P
variables s.t. Xn → X it holds
%(X) ≤ lim %(Xn ).
(1.5)
c) For any uniformly bounded sequence (Xn ) ⊂ L∞ (P ), Xn ↓ X implies
%(X) = lim %(Xn ).
(1.6)
(1.3)–(1.6) are in this general form due to Delbaen (2002).
In the following part of the paper we will mostly restrict to the representation by
the finitely additive measures even if extensions to a discussion of the Fatou-property are
possible.
2
Optimal allocation of risks and the related Pareto
equilibrium
In this section we establish that the Pareto equilibrium notion and its characterization
by Heath and Ku (2004) is naturally associated with properties of some risk measures
4
C. Burgert, L. Rüschendorf
which describe the optimal risk sharing problem in a market with n traders as introduced
in section 1 (optimistic view towards risks). The main result in this section shows that
the optimal risk sharing problem without constraints is well defined if and only if the
Pareto equilibrium condition holds. In the second part of this section we briefly indicate
an extension of these results to the case of incomplete markets.
2.1
Pareto equilibrium and related risk measures
Let (Ω, A, P ) be the underlying probability space and consider a market with n traders
with coherent risk measures %1 , . . . , %n , %i : L∞ (P ) → IR, acceptance sets A%i = {X ∈
L∞ (P ); %i (X) ≤ 0} and convex, σ(ba(P ), L∞ (P ))-closed representing scenario sets Pi ⊂
ba(P ), such that
%i (X) = sup EQ (−X),
1 ≤ i ≤ n.
Q∈Pi
One can consider the risk allocation problem as a game in the sense of game theory. From
this point of view Heath and Ku (2004) introduced and characterized the notion of Pareto
equilibrium for the allocation problem.
Definition 2.1 (Pareto equilibrium) A market model with risk measures %1 , . . . , %n is
in Pareto equilibrium, if
(E)
Xi ∈ L∞ (P ) with
n
X
Xi = 0 and %i (Xi ) ≤ 0,
1 ≤ i ≤ n,
i=1
implies %i (Xi ) = 0,
1 ≤ i ≤ n.
In a balance of supply and demand it is in equilibrium not possible to lower the risk
of some traders without increasing that of others. Vaguely one could say that there is no
arbitrage situation concerning risk. Equivalently, the equilibrium condition says that the
trivial decomposition 0 = 0+. . .+0 is a Pareto optimal decomposition of zero. We define as
optimal risk allocation
Pn problem the problem to determine the set of allocations (Xi )
of X that minimize
i=1 %i (Xi ). Under the Pareto equilibrium condition this infimum
is finite and the set of solutions coincides with the set of Pareto optimal allocations (see
Gerber (1979, page 89, 90)). Thus minimizing the total risk of allocations is equivalent
with determining Pareto optimal allocations under the equilibrium condition.
In this section we give a new derivation of the characterization of the Pareto equilibrium
condition (E) due to Heath and Ku (2004). We derive this result from properties of risk
measures which are naturally associated to this problem; in particular we make use of the
inf-convolution risk measure %b = %1 ∧ · · · ∧ %n and of a risk measure Ψ defined in terms of
the acceptance sets of %i . To derive this connection we first introduce a seemingly stronger
version of the Pareto equilibrium condition (E).
(SE)
Strong equilibrium
n
n
X
X
If Xi ∈ L∞ (P ) with
Xi = 0, then
%i (Xi ) ≥ 0.
i=1
i=1
It is immediate to see that (SE) ⇒ (E). Therefore, we call this condition strong equilibrium. But in fact both conditions are equivalent.
Proposition 2.2 The equilibria conditions (E) and (SE) are equivalent.
Pn
Pn
Proof: Assume that for some Xi ∈ L∞ (P ) with i=1PXi = 0 holds i=1 %i (Xi ) =: c < 0.
n
Then with ci := %i (Xi ) and Zi := Xi + ci − nc holds: i=1 Zi = 0 and
%i (Zi ) = %i (Xi ) − ci +
c
c
= < 0,
n
n
1 ≤ i ≤ n.
Allocation of risks and equilibrium in markets with finitely many traders
5
Thus we obtain a contradiction to (E).
2
Thus under the Pareto equilibrium condition (E) the sum of all risks in a balance
of supply and demand situation is nonnegative. The equilibrium condition (E) is closely
connected with the following risk measure Ψ defined by the acceptance set
n
³[
´
A%i
(2.1)
Ψ(X) := inf{m ∈ IR; X + m ∈ A}
(2.2)
A := cone
i=1
where
is the associated risk measure. All risk positions are acceptable if they are acceptable for any
of the traders in the market. Thus it seems natural that Ψ is connected with an optimistic
view towards risk (asPintroduced in section 1) and thus with the optimal risk sharing
n
problem to minimize i=1 %i (Xi ) over all allocations (Xi ). The equilibrium condition (E)
can be described in terms of the risk measure Ψ.
Proposition 2.3 Ψ is a coherent risk measure ⇔ Ψ(0) = 0
⇔ The equilibrium condition (E) holds.
Proof: The first equivalence is easy to check. For the second one assume that (E) holds.
Then by Proposition 2.2 also (SE) holds. By definition
(
)
n
X
Ψ(0) = inf m ∈ IR; ∃Xi ∈ A%i , 1 ≤ i ≤ n, m =
Xi .
i=1
By (SE) for any Xi ∈ A%i with
n
X
Xi − m = (X1 − m) +
i=1
n
X
Xi = 0
i=2
holds
%1 (X1 − m) +
n
X
%i (Xi ) =
i=2
n
X
%i (Xi ) + m ≥ 0,
i=1
Pn
i.e. m ≥ − i=1 %i (Xi ) ≥ 0, since %i (Xi ) ≤ 0. Thus we obtain
Pn Ψ(0) = 0.
Conversely, if Ψ(0) = 0 and if Xi ∈ A%i are in balance, i=1 Xi = 0. Then using that
Ψ ≤ %i ,
1 ≤ i ≤ n,
(2.3)
we obtain
0=Ψ
n
³X
i=1
≤
n
X
n
´ X
Xi ≤
Ψ(Xi )
i=1
%i (Xi ) ≤ 0.
i=1
This implies %i (Xi ) = 0, 1 ≤ i ≤ n, i.e. (E) holds.
2
As corollary we obtain a characterization of Ψ under the Pareto equilibrium condition
as largest coherent risk measure below min %i (X).
6
C. Burgert, L. Rüschendorf
Corollary 2.4 Let the market ((Ω, A, P ), %1 , . . . , %n ) with coherent risk measures %i be in
equilibrium (i.e. condition (E) holds). Then Ψ is the largest coherent risk measure % such
that
%(X) ≤ min{%i (X); 1 ≤ i ≤ n}.
(2.4)
Proof: Let % be a coherent risk measure % ≤ min1≤i≤n %i . Then X ∈ A%i implies that
X ∈ A% and thus
à n
!
[
A% ⊃ cone
A%i
(2.5)
i=1
This implies that % ≤ Ψ.
2
The equivalence of (E) and (SE) suggests to consider the infimal convolution %b =
%1 ∧ · · · ∧ %n defined by
( n
)
n
X
X
%b(X) := inf
%i (Xi );
Xi = X .
(2.6)
i=1
i=1
%b is the risk measure that describes the value of the optimal allocation of the total risk
X to the traders in the market, such that the sum of the allocated risks is minimal.
It is easy to check that %b satisfies all axioms of a coherent risk measure except possibly
the conditions %b(0) = 0.
Proposition 2.5
1) %b is a coherent risk measure ⇔ %b(0) = 0 ⇔ (SE) holds
2) Under the equilibrium condition (E) holds:
%b = Ψ.
(2.7)
Proof:
1) The first equivalence has been mentioned already. The second equivalence follows from
the definition of %b since
( n
)
n
X
X
%b(0) = inf
%i (Xi );
Xi = 0 = 0
i=1
i=1
is equivalent to
n
X
Xi = 0 ⇒
i=1
n
X
%i (Xi ) ≥ 0
i=1
i.e. to condition (SE).
2) Assume that (E) holds true. Then by 1) %b is a coherent risk measure
Pn and thus by
Corollary 2.4 we have %b ≤ Ψ. Conversely, for any decomposition X = i=1 Xi holds –
using Ψ ≤ min{%i } –
à n
!
n
n
X
X
X
%i (Xi ) ≥
Ψ(Xi ) ≥ Ψ
Xi = Ψ(X),
i=1
i.e. %b(X) ≥ Ψ(X).
i=1
i=1
2
Allocation of risks and equilibrium in markets with finitely many traders
7
Remark 2.6 As consequence of Proposition 2.5 we obtain that the optimal (unconstrained)
allocation problem makes only sense under the Pareto equilibrium condition (E). Without
condition (E) the optimal risk allocation problem leads to the inconsistency that %b(0) = −∞.
In particular without the Pareto equilibrium condition (E) it is not possible to determine
Pareto optimal allocations from the optimal allocation problem.
By the optimal proof of the general representation theorem of Delbaen (2002) (see (1.3))
there exists a σ(ba(P ), L∞ (P ))-closed representation set of scenarios P ⊂ ba(P ) such that
%(X) = supQ∈P EQ (−X). P can be chosen as the set of normed elements of the polar set
A0 of the acceptance set A = A% of %, i.e.
P = {Q ∈ ba(P ); Q1 = 1, EQ X ≥ 0, ∀X ∈ A} = P% .
(2.8)
Our aim is to describe the equilibrium condition (E) in terms of the scenario set P of Ψ.
The following proposition says that P contains exactly those scenario measures which are
common to all risk measures.
Proposition 2.7 Consider the market model with coherent risk measures %i and representation sets Pi ⊂ ba(P ), 1 ≤ i ≤ n and assume that the Pareto equilibrium condition (E)
holds: Then the representation set P = P%b = PΨ is nonempty and is given by
n
\
P=
Pi .
(2.9)
i=1
Proof: It holds
P = {Q ∈ ba(P ); EQ X ≥ 0, ∀X ∈ cone
n
³[
´
A%i }
i=1
= {Q ∈ ba(P ); ∀i ≤ n holds: EQ X ≥ 0, ∀X ∈ A%i }
n
\
=
{Q ∈ ba(P ); EQ X ≥ 0, ∀X ∈ A%i }
=
i=1
n
\
Pi .
2
i=1
As consequence of Propositions 2.5 and 2.7 we obtain the characterization result of
Heath and Ku (2004) for equilibrium in terms of the scenarios of the risk measures %i .
This result was stated in Heath and Ku (2004) for finite models Ω and finitely generated
scenario sets Pi .
Theorem 2.8 (Characterization of equilibrium) Consider the market with coherent
risk measures %1 , . . . , %n . Then the equilibrium condition (E) is equivalent to the condition
n
\
Pi 6= Ø,
(2.10)
i=1
i.e., there exists a scenario measure Q which is shared by all traders in the market.
Proof: If condition (E) holds, then by Proposition 2.7 we obtain P =
versely, if
n
T
i=1
Pi 6= Ø and Q ∈
%b(0) = inf
( n
X
i=1
%i (Xi );
i=1
n
T
i=1
n
X
i=1
n
T
Pi 6= Ø. Con-
Pi , then
)
Xi = 0
≥ inf
( n
X
i=1
EQ (−Xi );
n
X
i=1
)
Xi = 0
= 0.
8
C. Burgert, L. Rüschendorf
This implies that %b(0) = 0 and thus by Proposition 2.5 condition (E) holds.
2
Remark 2.9 a) In comparison to the proof of Heath and Ku (2004) who reduce the characterization problem to an application of the duality theorem of linear programming, our
proof is based on properties of the derived risk measures Ψ and %b which are naturally
associated to the problem.
b) The infimal convolution %b and the risk measure Ψ have already been introduced in the
literature and applied to the problem of risk transfer (see Barrieu and El Karoui (2004,
2005), and Delbaen (2000)). In these papers also related results on the acceptance set
and representation set of %b are given. In particular one finds there an investigation of
the Fatou-property of %b. The application of these risk measures to derive the equilibrium
characterization result is given in our paper for the first time.
c) Based on this paper (which has been circulated in march (2005)) we have extended in a
follow up paper (see Burgert and Rüschendorf (2006)) the results to the more general
case of convex risk measures. In this framework the Pareto equilibrium condition is no
longer equivalent to the fact that the optimal risk allocation problem is well defined, but
it is a stronger condition (see Proposition 3.3 in that paper). The intersection property
in Proposition 2.10 is in this case equivalent with the well posedness of the optimal
allocation problem. This equivalence is also derived independently in a paper by Jouini,
Schachermayer, and Touzi (2005) for the case of monetary utility functions the proof
there is based on an application of convex duality theory. Under this condition also
existence of Pareto optimal allocations is proved in that paper for law invariant monetary
utility functions.
d) An important early paper on the allocation problem is Deprez and Gerber (1985). In
that paper Deprez and Gerber characterize for convex premium prinicples (corresponding to monetary utility functions) Pareto optimal allocations (generalization of Borch’s
Theorem). Moreover, for the class of those premium principles, which are based on a
generalized principle of utility in that paper a no trade equilibrium premium notation is
introduced. The existence of a no trade equilibrium premium is equivalent to the Pareto
equilibrium notion in Definition 2.1 (see Theorem 16, 17 in Deprez and Gerber (1985)).
So their paper can be considered as an original source of the notion of convex risk measure and as an early relevant contribution to the allocation problem. We thank a reviewer
for a hint to this paper.
2.2
Incomplete models
The results of section 2.1 extend directly to incomplete models where trading of the i-th
trader is only possible in linear subspaces Mi ⊂ L∞ (P ), 1 ≤ i ≤ n. There are various
motivations for considering restricted classes of trading sets in the literature like restricted
resources, regulatory restrictions, technical restrictions. For some motivation and further
references in the context of the related assignment problem we refer to (Ramachandran
and Rüschendorf (2002)). We assume that the constants are contained
in the trading sets
Pn
IR ⊂ Mi and define risk measures on the trading space M := i=1 Mi , by defining
Ai := {Xi ∈ Mi ; %i (Xi ) ≤ 0}
à n
!
[
AM := cone
Ai
(2.11)
(2.12)
i=1
For X ∈ M we introduce the modified version of the risk measure Ψ defined by
ΨM (X) := inf{m ∈ IR; m + X ∈ AM },
( n
)
n
X
X
%bM (X) := inf
%i (Xi ); Xi ∈ Mi ,
Xi = X .
i=1
i=1
(2.13)
(2.14)
Allocation of risks and equilibrium in markets with finitely many traders
9
Then we obtain as in section 2.1:
ΨM is a coherent risk measure on M ⇔ ΨM (0) = 0
(2.15)
%bM is a coherent risk measure on M ⇔ %bM (0) = 0.
(2.16)
The Pareto equilibrium condition for the incomplete market case is defined by
n
X
(EM ) Xi ∈ Mi with
Xi = 0 and %i (Xi ) ≤ 0 for all i
(2.17)
i=1
implies %i (Xi ) = 0 for all i.
The Pareto equilibrium condition(EM ) is equivalent to ΨM (0) = 0.
(2.18)
The corresponding strong equilibrium condition is defined by
(SEM ) Xi ∈ Mi and
X
i=1
Xi = 0 implies
n
X
%i (Xi ) ≥ 0.
(2.19)
i=1
The strong equilibrium condition (SEM ) is equivalent to %bM (0) = 0.
(2.20)
As consequences we obtain for the incomplete case the following conclusions in a similar
way as in section 2.1 for the complete case.
Proposition 2.10 Under the Pareto equilibrium condition (EM ) holds for the incomplete
model:
1) ΨM is the largest coherent risk measure on M such that ΨM/Mi ≤ %i/Mi , 1 ≤ i ≤ n,
where ΨM/Mi and %M/Mi denote the restrictions of ΨM and %i to the trading set Mi .
2) The risk measures %bM and ΨM are identical. The corresponding scenario sets are given
by
s
P%bM = PΨM = {Q ∈ ba(P ); Q/Mi ∈ Pi/Mi , 1 ≤ i ≤ n}
(2.21)
s
where ba(P ) is the set of signed finitely additive measures absolutely continuous w.r.t.
P.
The restriction to subspaces Mi does in general not imply positivity of the representing
measures – as in the complete market. In consequence we obtain as in section 2.1 an
extension of the characterization of Pareto equilibria in incomplete models which for finite
models Ω was given in Heath and Ku (2004).
Theorem 2.11 In the incomplete market case the equilibrium condition (EM ) is equivalent
to the following condition:
s
∃ Qi ∈ Pi and ∃ Q ∈ ba(P ) such that Q/Mi = Qi/Mi , 1 ≤ i ≤ n,
(2.22)
i.e., there exists a common scenario Q on the trading spaces Mi . Q is a signed measure
which is positive on Mi .
10
3
C. Burgert, L. Rüschendorf
The optimal risk allocation problem
The allocation results for the infimal convolution risk measure %b in section 2.1 leave open
the question how to allocate optimally risk when the equilibrium condition does not hold.
It is of interest to note that in many situations of practical relevance the equilibrium
condition (E) does not hold. The most simple example of this type is the case where
%i (X) = EQi (−X), 1 ≤ i ≤ n, where Qi are P continuous probability measures, which
represent the view towards risk of the i-th trader. If not all views Qi are identical, then
the equilibrium condition does not hold. Generally one can say that (E) does not hold if
the views of the traders towards risk are in some sense too different. In the other direction
the equilibrium condition does hold if
%i (Xi ) ≥ E(−Xi ),
1 ≤ i ≤ n,
the expectation w.r.t. P , because, then for any decomposition (Xi ) of zero
holds
n
X
i=1
%i (Xi ) ≥ E
(3.1)
Pn
i=1
Xi = 0
n
X
(−Xi ) = 0
i=1
and thus %̂(0) = 0. In particular, for all law invariant convex risk measures %i , i.e. where
%i (X) only depends on the law of X w.r.t. P , condition (3.1) holds. Thus the equilibrium
conditon holds, if %1 , . . . , %n are all law invariant.
Thus it is a problem of interest, to modify the risk allocation problem so that it makes
sense also in case the equilibrium condition does not hold. The main idea to deal with this
situation is, to restrict the class of allowed allocations in order to admit no ‘pathological’
allocations. This is similar to the restriction to admissible strategies in portfolio theory
in order to avoid that effects like doubling strategies may occur in risk allocation. Some
types of restrictions have been introduced in the insurance literature (see the introduction
of this paper). Our aim is to introduce restrictions as weak as possible which still yield a
senseful version of the optimal allocation problem and thus allow to determine w.r.t. this
class Pareto optimal allocations resp. risk sharing strategies. We introduce at first some
stricter boundedness assumptions on the allocations and show later (in Remark 3.6) that
a weakened form of these restrictions leads to the same optimal allocations.
Pn
Define a decomposition X = i=1 Xi to be admissible if
X(ω) ≥ 0 implies that 0 ≤ Xi (ω) ≤ X(ω) and X(ω) ≤ 0 implies that X(ω) ≤ Xi (ω) ≤ 0.
Thus for X(ω) ≥ 0 any trader has to take some nonnegative share of the risk, while
for X(ω) ≤ 0 only non positive shares are allowed. This restriction does not permit uncontrolled ‘borrowing’ as e.g. unrestricted puts and calls and as a consequence prevents
risk arbitrage. We will see in Theorem 3.5 that these restrictions are mathematically well
justified. Economically the introduced restrictions can be motivated by a game theoretic
argument considering two kind of players, the traders on one side and some regulatory
agent on the other side, whose intension is to prevent ‘artificial’ risk reduction on the side
of the traders, which one might call ‘risk arbitrage’. The means to do this is to put suitable
restrictions on the class of allocations. In the following we will show that the restrictions
introduced have some kind of minimax character. They are the ‘mildest’ form of restrictions which prevent risk arbitarge. From an applied point of view restrictions as introduced
here can be seen as introducing a protection against artificial risk reduction in the group
of traders. This artificial risk reduction is possible if the risk measures used by the traders
are too different which is described mathematically by the property that the equilibrium
condition does not hold. The idea of this restriction is thus similar to the corresponding
restrictions on strategies in portfolio theory.
Let A(X) denote the set of admissible decompositions (Xi ) of X. Our definition puts
constraints on the compensation structure with respect to the losses. In comparison to
Allocation of risks and equilibrium in markets with finitely many traders
11
Gerber’s constraints (see section 1) our constraints depend on the risk X and are not fixed
upper or lower bounds on the Xi . We will show that this class of restrictions avoids the
unwanted pathological effects and leads to an allocation problem which can not be improved
without obtaining pathologies. It will turn out also that we can weaken boundedness of
the restrictions somewhat without changing the allocation problem (see Remark 3.6).
We define the admissible infimal convolution %∗ by
%∗ (X) := inf
( n
X
)
%i (Xi ); (Xi ) ∈ A(X) .
(3.2)
i=1
%∗ describes the optimal total risk w.r.t. all admissible allocations. Obviously, we have
%∗ ≤ min %i . %∗ is a subadditive, homogeneous monotone risk functional – in particular
%∗ (0) = 0 – but %∗ is not translation invariant in general. From the definition of %∗ we only
obtain
( n
)
n
X
X
%∗ (1) = inf
%i (Xi ); 0 ≤ Xi ,
Xi = 1
(3.3)
i=1
i=1
≤ %i (1) = −1.
In the following theorem we derive an essential simplified dual representation of %∗ in
terms of the scenario measures Pi of %i . To obtain this representation we make use of an
alternative description of the admissible decompositions like in multiple decision problems:
(
(ϕi X); 0 ≤ ϕi ≤ 1,
A(X) =
n
X
)
ϕi = 1 .
(3.4)
i=1
For Pi ∈ Pi let P1 ∧ · · · ∧ Pn denote the lattice infimum of (Pi ) in the lattice ba(P ) and
let P1 ∨ · · · ∨ Pn denote the lattice supremum of (Pi ) in ba(P ). A careful introduction to
finitely additive measures and their lattice structure is given in Rao and Rao (1983), see
in particular Theorem 2.2.1. In the case that Pi are probability measures with densities fi
w.r.t. µ then P1 ∧ · · · ∧ Pn resp. P1 ∨ · · · ∨ Pn have densities min{fi } resp. max{fi } w.r.t.
µ. The admissible infimal convolution %∗ admits the following useful dual representation
of %∗ using the lattice infima and suprema. This representation simplifies essentially the
calculation of %∗ and is useful also in the following.
Theorem 3.1 Let %j = %Pj be coherent risk measures.
1) The admissible infimal convolution %∗ has the dual representation
(Z
%∗ (X) = sup
X− d
V
j
Z
Pj −
½
2)
A%∗ =
X+ d
Z
X ∈ L∞ (P ); such that
X− d
W
j
V
)
Pj ; Pj ∈ Pj , 1 ≤ j ≤ n .
Z
Pj ≤
X+ d
W
¾
Pj for all Pj ∈ Pj
Proof:
1) For Pi ∈ Pi , 1 ≤ i ≤ n and Y := −X ∈ L∞ (P ) holds
aP1 ,...,Pn (Y ) := inf
( n Z
X
i=1
ϕi Y dPi ; 0 ≤ ϕi ,
n
X
i=1
)
ϕi = 1
(3.5)
. (3.6)
12
C. Burgert, L. Rüschendorf
has a solution (ϕ∗i ) and if Y (ω) > 0, then {ϕ∗i > 0} ⊂ {Pi =
Wn
then {ϕ∗i > 0} ⊂ {Pi = j=1 Pj }. Thus
Z
aP1 ,...,Pn (Y ) =
j=1
Y ≥0
Z
=
Y+ d
V
Z
=
Z
n
V
Yd
X− d
Pj +
V
j=1
Y <0
Z
Pj −
Y− d
W
Z
Pj −
n
W
Yd
X+ d
Vn
j=1
Pj } and if Y (ω) < 0,
Pj
Pj
W
Pj .
Therefore, we obtain
%∗ (X) = inf
(ϕi )
X
%i (ϕi X) = inf
(ϕi )
i
"
= inf −
(ϕi )
i=1
= − sup
(ϕi )
Z
n
X
X
i
inf
Pi ∈Pi
Z
X
i
sup
Pi ∈Pi
(−ϕi X)dPi
#
ϕi XdPi
Z
inf
Pi ∈Pi
ϕi XdPi
We now apply a useful and general version of the minimax theorem which can be found
in Müller (1971).
Minimax Theorem: Let f : A × B → IR, A, B 6= Ø be a game of concave-convex type,
i.e.
1) ∀ b1 , b2 ∈ B, α ∈ [0, 1] there exists a b ∈ B such that for all a ∈ A holds
f (a, b) ≤ (1 − α)f (a, b1 ) + αf (a, b2 ).
2) ∀ a1 , a2 ∈ A, α ∈ [0, 1] there exists an a ∈ A such that for all b ∈ B holds
f (a, b) ≥ (1 − α)f (a1 , b) + αf (a2 , b).
If f < ∞ and for some topology τ on A holds A is τ -compact and ∀ b ∈ B,
f (·, b) : A → IR is upper semicontinuous, then
inf sup f (a, b) = sup inf f (a, b).
b∈B a∈A
(3.7)
a∈A b∈B
P
We choose A = {(ϕi ); 0 ≤ ϕi , ϕi =P1}, Rwhich is compact in weak∗-topology,
n
B = P1 × · · · × Pn and f ((ϕi ), (Pi )) =
ϕi XdPi . By linearity of f and coni=1
vexity of Pi and A the conditions of the minimax theorem are fulfilled and we obtain
from the first part of the proof
%∗ (X) = − inf sup
Pi ∈Pi (ϕi )
µZ
= sup
Pi ∈Pi
2) follows from 1).
XZ
µZ
ϕi XdPi = − inf
i
V
X− d Pi −
Pi ∈Pi
Z
W
X+ d Pi −
Z
V
X− d Pi
¶
¶
W
X + d Pi .
2
An interesting consequence of Theorem 3.1 is the following characterization of the
Pareto equilibrium condition (E) in terms of %∗ .
Allocation of risks and equilibrium in markets with finitely many traders
13
Proposition 3.2 Let %i be coherent risk measures and let %∗ denote the admissible infimal
convolution, then it holds:
%∗ is a coherent measure ⇔ %∗ (1) = −1
⇔ The Pareto equilibrium condition (E) holds.
Under (E) we have %∗ = %b = Ψ.
Proof: The first equivalence is obvious since the condition %∗ (1) = −1 implies translation
invariance of %∗ . By Theorem 3.1
(
)
W
%∗ (1) = sup −| Pj |; Pj ∈ Pj
j
n W
o
= − inf | Pj |; Pj ∈ Pj .
j
Thus %∗ (1) = −1 if and only if there exists a common scenario measure Q ∈
by Theorem 2.8 is equivalent to the Pareto equilibrium condition (E).
Tn
i=1
Pi which
2
Remark 3.3 The condition %∗ (1) = −1 has the following interpretation. The traders in the
market try to allocate their risk in the best possible way which leads to a total risk %∗ (1) ≤
−1 for any risk measures %i . On the other hand from a regulatory point of view the risk
measures should be chosen by the traders in a cautious way in order not to underestimate
the whole risk. This game theoretic consideration suggests that in order to obtain that the
optimal admissible total risk is reasonable i.e. in our context is a coherent risk measure
one might expect that the condition %∗ (1) = −1 should hold. This idea is confirmed by
Proposition 3.2.
In the general case we modify %∗ to obtain a coherent risk measure which we call
coherent admissible infimal convolution.
Definition 3.4 (Coherent admissible infimal convolution) We define the coherent
admissible infimal convolution risk measure %b∗ by
%b∗ (X) := inf{m ∈ IR; X + m ∈ A}
= inf{m ∈ IR; %∗ (X + m) ≤ 0}
(3.8)
where A = A%∗ = {X; %∗ (X) ≤ 0}.
From the definition %b∗ is a coherent risk measure with acceptance set A%b∗ ⊃ A%∗ = A
and
%b∗ ≤ %∗ ≤ min %i .
(3.9)
The following theorem says that %b∗ is the largest coherent risk measure % ≤ min %i . This
is a justification for our choice of restrictions on the class of decompositions. Essentially less
severe restrictions do not lead to a coherent allocation problem and thus admit pathological
decompositions.
Theorem 3.5 Let %1 , . . . , %n be coherent risk measures, then:
1) Under condition (E) holds
%b∗ = %b = Ψ
2) %b∗ is the largest coherent risk measure % such that % ≤ mini %i .
14
C. Burgert, L. Rüschendorf
Proof:
1) %b∗ is the largest coherent risk measure % with % ≤ %∗ . Under condition (E) %∗ is a
coherent risk measure by Proposition 3.2. Therefore %b∗ = %∗ = %b = Ψ.
2) If % is
% ≤ min %i and if X ∈ L∞ (P ) has a decomposition
Pan coherent risk measure
P
X = i=1 ϕi X, 0 ≤ ϕi , ϕi = 1, then
X
X
%(X) ≤
%(ϕi X) ≤
%i (ϕi X).
i
i
This implies that %(X) ≤ %∗ (X) and thus
A% = {X ∈ L∞ (P ); %(X) ≤ 0}
⊃ {X ∈ L∞ (P ); %∗ (X) ≤ 0}
= A%∗
As consequence we obtain
%(X) = inf{m ∈ IR; X + m ∈ A% }
≤ inf{m ∈ IR; X + m ∈ A%∗ } = %b∗ (X).
2
Remark 3.6 (Enlarged class of admissible decompositions) It is possible to enlarge
the class of admissible decomposition sets without changing the total risk measure. Define
(
)
n
X
e
A(X)
= (Xi );
Xi = X and |Xi | ≤ |X| .
(3.10)
i=1
e has an equivalent description by
Then A
n
o
X
e
A(X)
= (ϕi X); |ϕi | ≤ 1,
ϕi = 1 .
(3.11)
Define %̃ as the corresponding infimal convolution
o
nX
e
%i (Xi ); (Xi ) ∈ A(X)
.
%e(X) = inf
(3.12)
Then %e is a subadditive, homogeneous, monotone risk functional. %e is a coherent risk measure if and only if %e(1) = −1. We obtain a similar explicit representation result as that for
%∗ in Theorem 3.1. Introducing
%b
e(X) = inf{m ∈ IR; %e(X + m) ≤ 0}
(3.13)
then %b
e is a coherent risk measure. Further for any coherent risk measure % ≤ min %i and
e
any X ∈ L∞ (P ) with decomposition (Xi ) ∈ A(X)
holds
X
X
%(X) ≤
%(Xi ) ≤
%i (Xi ) and thus % ≤ %b
e.
(3.14)
i
i
This implies by Theorem 3.5 that
%b
e = %b∗ .
(3.15)
e
Thus also with the enlarged class A(X)
of admissible decompositions we obtain the same
optimal risk decomposition rule.
Remark 3.7 Theorem 3.5 justifies the restriction to admissible decompositions for a general formulation of the optimal risk allocation problem. The coherent admissible infimal
convolution %b∗ is in the general case (without equilibrium condition) the relevant coherent
risk measure for the allocation problem describing the total ‘intrinsic’ risk of X after an
optimal admissible allocation to the traders.
Allocation of risks and equilibrium in markets with finitely many traders
4
15
Allocation of risks and cautious risk attitude
In this section we consider in a market with n traders with risk measures %1 , . . . , %n the
allocation problem from a regulatory agent’s point of view (cautious risk attitude). The
aim is to apply cautious risk measurement methods in order to meet worst case situations.
So in contrast to sections 2, 3 we do not consider the problem of optimal risk sharing
but take the view of a regulator and consider non cooperating traders where the total risk
X is distributed in some way to the traders. The regulator would like to take care for
the worst case of allocation of risk and the corresponding necessary capital to meet this
situation. For the cautious risk attitude a first natural choice of risk measure is
%max (X) := max %i (X)
(4.1)
i
(see Delbaen (2000), Föllmer and Schied (2004)). It is easy to check that %max is a coherent
risk measure. The acceptance set is given by
A%max = {X ∈ L∞ (P ); %max (X) ≤ 0}
= {X ∈ L∞ (P ); %i (X) ≤ 0, ∀ i}
n
\
=
A%i .
(4.2)
i=1
The representing scenario set is given by
P%max = conv
n
³[
´
Pi
(4.3)
i=1
since
X ∈ A%max ⇔ X ∈ A%i , 1 ≤ i ≤ n
⇔ ∀ Qi ∈ Pi : EQi X ≥ 0, 1 ≤ i ≤ n
n
³[
´
⇔ ∀ Q ∈ conv
Pi : EQ X ≥ 0.
i=1
%max is obviously the smallest coherent risk measure majorizing %i , 1 ≤ i ≤ n. No equilibrium condition is connected with the application of %max .
Remark 4.1 (Incomplete markets) %max can also be applied in the incomplete market situation M ⊂ L∞ (P ), M a linear subspace and with the restrictions %i/M . A result
from the theory of convex cones (cf. Fuchssteiner and Lusky (1981)) implies
that any
Pn
µ ∈ ba(P/M ) with µ ≤ %max/M has a representation of the form µ = k=1 λk µk with
λk ≥ 0 and µk ≤ %k , 1 ≤ k ≤ n, µk ∈ ba(P/M ). Thus also in the incomplete market case
we get a representation of %max/M by
(
M
Pmax
=
s
µ ∈ ba(P ); µ/M ∈ conv
n
³[
´
Pi/M
)
.
i=1
A more cautious total risk measure arises when the risk is allocated to the traders in
the most unfavorable way. This leads to
Definition 4.2 (Supremal convolution) For coherent risk measures %1 , . . . , %n define
the supremal convolution τb := %1 ∨ · · · ∨ %n by
( n
)
n
X
X
τb(X) := sup
%i (Xi );
Xi = X .
(4.4)
i=1
i=1
16
C. Burgert, L. Rüschendorf
τb satisfies all conditions of a coherent risk measure except possibly the condition τb(0) =
0. τb(0) can be interpreted as a possible hidden risk position in the market when decomposing
the zero position in an unfavorable way to the traders.
To investigate this we introduce as in section 2.1 Pareto equilibrium conditions:
(Eτ )
n
X
Xi = 0 and %i (Xi ) ≥ 0, ∀ i implies %i (Xi ) = 0, ∀ i.
(4.5)
i=1
The corresponding strengthened version of (Eτ ) is
(SEτ )
n
X
n
X
Xi = 0 implies
i=1
%i (Xi ) ≤ 0.
(4.6)
i=1
As in Proposition 2.2 we find:
Proposition 4.3 1) The equilibrium conditions (Eτ ) and (SEτ ) are equivalent.
2) τb is a coherent risk measure ⇔ τb(0) = 0
⇔ The equilibrium condition (Eτ ) holds.
In the following proposition we establish that – in spite of the formal similarity of
condition (Eτ ) to (E) – except in some exceptional cases the equilibrium condition (Eτ )
does not hold.
Proposition 4.4 Consider coherent risk measures %i = %Pi . Then the equilibrium condition (Eτ ) holds if and only if P1 = P2 = · · · = Pn and |Pi | = 1.
Proof: We consider at first the case n = 2. Then (Eτ ) is equivalent to:
∀X ∈ L∞ (P ) : sup EQ X − inf EQ X ≤ 0.
Q∈P2
Q∈P1
This is further equivalent to:
∀X ∈ L∞ (P ) holds :
sup EQ X ≤ inf EQ X
Q∈P1
Q∈P2
and
sup EQ X ≤ inf EQ X
Q∈P2
Q∈P1
considering X as well as −X.
Thus we obtain equivalence to
sup EQ X ≤ inf EQ X
Q∈P1
Q∈P2
≤ sup EQ X ≤ inf EQ X,
Q∈P2
Q∈P1
∀X ∈ L∞ (P )
or equivalently
sup EQ X = inf EQ X
Q∈P1
(4.7)
Q∈P1
= inf EQ X = sup EQ X,
Q∈P2
∀X ∈ L∞ (P ).
Q∈P2
(4.7) again is equivalent to
P1 = P2 and |P1 | = |P2 | = 1.
In the case n ≥ 2 we obtain from the recursive structure of the supremal convolution
τb = (%1 ∨ · · · ∨ %n−1 ) ∨ %n .
(4.8)
Thus the result for the case n = 2 implies:
P%n = P%1 ∨···∨%n−1 = P%n−1 = · · · = P%1 and |P%i | = 1
is equivalent to the equilibrium condition (Eτ ).
(4.9)
2
Allocation of risks and equilibrium in markets with finitely many traders
17
Remark 4.5 (Pathological risk decompositions) As consequence of Proposition 4.4
we find that the equilibrium condition (Eτ ) is only satisfied in the exceptional case that
P1 = P2 = · · · = Pn = {P1 }. In all other situations it follows from the homogeneity of τb
that
τb(0) = ∞.
(4.10)
Thus for any K > 0 there exist pathological risk decompositions (Xi ) of 0
0=
n
X
n
X
Xi with
i=1
%i (Xi ) ≥ K.
(4.11)
i=1
Thus in the marketP
there may be hidden allocations of the risk 0 to the traders with arbitrary
large sum of risks
%i (Xi ) if any decomposition of X is taken into consideration.
As consequence of Proposition 4.4 it is necessary to prevent the regulatory agent to
be too strict since this would lead to ‘artificial’ worst case scenarios with too high risk. It
seems natural to restrict the class of admissible decompositions as in section 3. Let
½
¾
n
X
A(X) := (Xi ) = (ϕi X); 0 ≤ ϕi ≤ 1,
ϕi = 1
(4.12)
i=1
denote the class of admissible decompositions and define the admissible supremal convolution
½X
¾
n
τ ∗ (X) := sup
%i (Xi ); (Xi ) ∈ A(X) .
(4.13)
i=1
Then τ ∗ is a subadditive homogeneous monotone risk functional (with τ ∗ (0) = 0) but τ ∗
is not translation invariant in general. Obviously
%max ≤ τ ∗ ≤ τb.
(4.14)
As in Theorem 3.1, τ ∗ can be calculated explicitly in terms of the representation scenarios
Pi of %i .
Theorem 4.6 Let %i = %Pi be coherent risk measures and τ ∗ the corresponding admissible
supremal convolution. ½Z
Then
¾
Z
W
V
τ ∗ (X) = sup
X− d Pi − X+ d Pi ; Pi ∈ Pi , 1 ≤ i ≤ n
1)
i
i
½
¾
Z
Z
W
V
∞
Aτ ∗ = X ∈ L (P ); such that
X− d Pi ≤ X+ d Pi , ∀Pi ∈ Pi .
2)
i
i
Proof:
1) As in the proof of Theorem 3.1 we obtain
X
τ ∗ (X) = sup
%i (ϕi X)
(ϕi )
= sup
(ϕi )
=
i
X
i
sup
sup EPi ϕi Y, with Y := −X
Pi ∈Pi
sup
XZ
(Pi )∈(Pi ) (ϕi )
µZ
= sup
(Pi )
= sup
(Pi )
2) follows from 1).
Y+ d
ϕi Y dPi
i
W
µZ
X− d
i
Z
Pi −
W
i
Y− d
Z
Pi −
V
i
X+ d
¶
Pi
V
i
¶
Pi .
2
18
C. Burgert, L. Rüschendorf
Remark 4.7 a) As consequence of Theorem 4.6 we see that
n¯ V ¯
o
τ ∗ (1) = − inf ¯ Pi ¯; Pi ∈ Pi
(4.15)
and thus
¯V ¯
τ ∗ (1) = −1 ⇔ ¯ Pi ¯ = 1, ∀Pi ∈ Pi
⇔ P1 = P2 = · · · = Pn and |Pi | = 1
(4.16)
⇔ The Pareto equilibrium condition (Eτ ) holds.
b) Enlarged class of admissible strategies. As in section 3 we can enlarge the class of
Pn
e
admissible decompositions to A(X)
= {(Xi );
i=1 Xi = X and |Xi | ≤ |X|}. We then
get a corresponding risk functional
nX
o
e
τe(X) := sup
%i (Xi ); (Xi ) ∈ A(X)
.
(4.17)
Similarly to Theorem 4.6 we obtain an explicit representation of τe by
½Z
Z
W
V
τe(X) = sup
X− d Pi − X+ d Pi
(4.18)
µZ
¶
¾
Z
W
V
n−1
X + d P i − X + d P i ; P i ∈ Pi , 1 ≤ i ≤ n .
+
2
As consequence we obtain that in contrast to the infimum case in section 2.2 the enlargement of admissible strategies leads to a different risk measure
τ ∗ (X) ≤ τe(X)
(4.19)
and equality holds only under the Pareto equilibrium condition (Eτ ).
Definition 4.8 (Coherent admissible supremal convolution) The coherent admissible supremal convolution risk measure τb∗ is defined by
τb∗ (X) = inf{m ∈ IR; X + m ∈ Aτ ∗ } = inf{m ∈ IR; τ ∗ (X + m) ≤ 0},
(4.20)
where Aτ ∗ = {X ∈ L∞ (P ); τ ∗ (X) ≤ 0}.
In the following proposition we characterize τ̂ ∗ by a maximality property.
Proposition 4.9 1) The coherent admissible supremal convolution risk measure τb∗ is the
largest coherent risk measure % such that
τ ∗ ≥ % ≥ %max .
2) If the %i have the Fatou-property, then Aτb∗ = Aτ ∗ and the scenarios representation set
P ∗ of τb∗ is given by
L1 (P )
P ∗ = conv Q
,
n
V
W o
Q
where Q = |Q|
∈ M 1 (P ); ∃Pi ∈ Pi , 1 ≤ i ≤ n, with
Pi ≤ Q ≤ Pi and M 1 (P )
denotes the P -continuous probability measures.
Proof:
1) τb∗ is the largest coherent risk measure % with % ≤ τ ∗ .
Allocation of risks and equilibrium in markets with finitely many traders
19
2) By the bipolar theorem and the representation of τ ∗ in Theorem 4.6 it is to show that
Z
Z
W
V
X− d Pi ≤ X+ d Pi , ∀Pi ∈ Pi
is equivalent to
Z
Z
X− dQ ≤ X+ dQ,
∀Q ∈ P ∗ .
V
W
If Q ∈ P ∗ and Pi ≤ αQ ≤ Pi with some positive constant α then
Z
Z
Z
Z
W
V
α X− dQ ≤ X− d Pi ≤ X+ d Pi ≤ α X+ dQ.
For the other direction let Pi ∈ Pi , 1 ≤ i ≤ n and define Q := 1{X≥0}
Q
Then |Q|
∈ P ∗ and
Z
X− d
W
Z
Pi =
Z
X− dQ ≤
Z
X+ dQ =
X+ d
V
V
Pi +1{X<0}
W
Pi .
Pi .
Thus P ∗ is the representation set of τb∗ since Aτb∗ = Aτ ∗ = (P ∗ )0 and thus
conv({0} ∪ P ∗ ) = (Aτb∗ )0 .
2
Conclusion: The classical risk allocation (risk sharing) problem is only well defined if the
view towards risks of the traders is not too different, in mathematical terms if the Pareto
equilibrium condition holds. Thus it is of interest to extend the allocation problem in a
senseful way to the general case. An effective way for the formulation of a general allocation
problem is to introduce restrictions on the class of admissible allocations in a suitable way.
This is done in this paper for the risk sharing problem (under ‘optimistic risk attitude’) and
also under a ‘cautious risk attitude’ (regulatory agents view). As consequence we obtain
new relevant coherent risk measures for allocation problens. A simplified dual representation for these intrinsic risk measures is derived. The choice of the restriction conditions is
justified by the fact that essentially less restrictive conditions would allow ‘pathological’
allocations (‘risk arbitrage’) and in particular would not allow to obtain Pareto optimal
allocations.
Acknowledgement: The authors thank a reviewer for several useful hints and remarks
on this paper.
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Department of Mathematical Stochastics
University of Freiburg
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